Question

How do you simplify \[\dfrac{20+40x}{20x}\]?

Answer 1

Hint: This question is from the topic of algebra. In this question, we are going to simplify the given term. Using the prime factorization method, we will first take out the common factor from the numerator. After that, we will take out the factor from the denominator. After that, we will cancel them out and solve the further equation by splitting the terms that are in addition and get the answer.

Complete step-by-step solution:
Let us solve this question.
In this question, we have asked to simplify the term\[\dfrac{20+40x}{20x}\]. That means we will solve the term \[\dfrac{20+40x}{20x}\] and make it simple.
Now, we will find out the prime factors of the term 20 and 40.
Finding for prime factorization of 40:
\[\begin{align}
  & 2\left| \!{\underline {\,
  40 \,}} \right. \\
 & 2\left| \!{\underline {\,
  20 \,}} \right. \\
 & 2\left| \!{\underline {\,
  10 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
So, prime factorization of 40 will be
\[40=2\times 2\times 2\times 5\times 1\]
Finding for prime factorization of 20:
\[\begin{align}
  & 2\left| \!{\underline {\,
  20 \,}} \right. \\
 & 2\left| \!{\underline {\,
  10 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
So, prime factorization of 20 will be
\[20=2\times 2\times 5\times 1\]
Now, we can write equation \[\dfrac{20+40x}{20x}\] as
\[\dfrac{20+40x}{20x}=\dfrac{2\times 2\times 5\times 1+2\times 2\times 2\times 5\times 1\times x}{2\times 2\times 5\times 1\times x}\]
The above equation can also be written as
\[\Rightarrow \dfrac{20+40x}{20x}=\dfrac{2\times 2\times 5+2\times 2\times 2\times 5\times x}{2\times 2\times 5\times x}\]
Now, we can take out \[2\times 2\times 5\] as a common factor from the numerator.
After taking the common factor \[2\times 2\times 5\] from the above equation, we can write
\[\Rightarrow \dfrac{20+40x}{20x}=\dfrac{2\times 2\times 5\times \left( 1+2\times x \right)}{2\times 2\times 5\times x}\]
Now, we can see that there is a factor of \[2\times 2\times 5\] in the denominator, we can write
\[\Rightarrow \dfrac{20+40x}{20x}=\dfrac{2\times 2\times 5\times \left( 1+2\times x \right)}{2\times 2\times 5\times \left( x \right)}\]
Now, we can see that there is a common factor of the term \[2\times 2\times 5\] in both numerator and denominator, then we will cancel them out. After cancelling them, we can write
\[\Rightarrow \dfrac{20+40x}{20x}=\dfrac{\left( 1+2\times x \right)}{\left( x \right)}\]
The above equation can also be also be written as
\[\Rightarrow \dfrac{20+40x}{20x}=\dfrac{1+2x}{x}\]
Now, we will split the terms.
We can write the above equation after splitting as
\[\Rightarrow \dfrac{20+40x}{20x}=\dfrac{1}{x}+\dfrac{2x}{x}\]
Now, we can see that there is an x which is common in both numerator and denominator. After cancelling x, we can write the above equation as
\[\Rightarrow \dfrac{20+40x}{20x}=\dfrac{1}{x}+2\]
Hence, we have simplified the term\[\dfrac{20+40x}{20x}\]. The simplified term is \[\dfrac{1}{x}+2\].
Here, we cannot take x as zero. Because after putting the value of x as zero, we will get an undefined value. So, the answer is the same but x cannot be zero.

Note: We should have a better knowledge in the topic of algebra for solving this type of question. Always remember how to find out the prime factorization of any number. This can be very helpful in some questions. We can solve this question by an alternate method.
The equation is \[\dfrac{20+40x}{20x}\]
As we know that if we multiply the number 20 by 2, then we get the multiplied number as 40.
So, we can write the above equation as
\[\Rightarrow \dfrac{20+40x}{20x}=\dfrac{20+20\times 2\times x}{20x}\]
Now, we can see that 20 is a common factor in the numerator, so we can write
\[\Rightarrow \dfrac{20+40x}{20x}=\dfrac{20\left( 1+2x \right)}{20x}\]
Now, the number 20 will be cancelled out because it is in both numerator and denominator.
So, we can write the above equation as
\[\Rightarrow \dfrac{20+40x}{20x}=\dfrac{1+2x}{x}\]
We can write the above equation by splitting the terms as
\[\Rightarrow \dfrac{20+40x}{20x}=\dfrac{1}{x}+\dfrac{2x}{x}\]
Now, from here we have found in the above that
\[\Rightarrow \dfrac{20+40x}{20x}=\dfrac{1}{x}+2\]
So, we have got the same answer from this method.